A note on the asymptotic behaviour of the distance of the knth nearest neighbour
Hannelore Liero · Statistics · 1993
Summary. We consider the asymptotic behaviour of the distance Rn(t) between a point t(d)Rdand the knth nearest neighbour of t among a sample X1,…, Xnof i.i.d. random vectors, Xi Rd, for n(d)(d). It is shown, that under appropriate conditions on the underlying distribution of XiRn(t) is asymptotically equivalent to a nonrandom sequence an(t). This equivalence relation holds uniformly in t, with probability one and with a certain rate of convergence. The result is useful for the study of nonparametric density and regression estimates of nearest neighbour type.